Cluster spin glass correlations and dynamics in Zn$_{0.5}$Mn$_{0.5}$Te

We present a magnetometry, muon spin relaxation ($\mu$SR), and neutron scattering study of the insulating face-centered-cubic spin glass Zn$_{0.5}$Mn$_{0.5}$Te. The magnetometry and $\mu$SR results confirm a spin freezing transition around $T_f \approx 23$ K, with the spin fluctuation rate decreasing gradually and somewhat inhomogeneously through the sample volume as the temperature decreases toward $T_f$. Characteristic spin correlation times well above $T_f$ are on the order of 10$^{-10}$ s, in line with expectations for a cluster spin glass. Using magnetic pair distribution function (mPDF) analysis and reverse Monte Carlo (RMC) modeling of the magnetic diffuse neutron scattering data, we show that the spin-glass ground state consists of clusters of spins exhibiting short-range-ordered type-III antiferromagnetic correlations, with a locally ordered moment of 3.1(1) $\mu_{\mathrm{B}}$ between nearest-neighbor spins. The type-III correlations decay exponentially as a function of spin separation distance with a correlation length of approximately 5 \AA. The diffuse magnetic scattering and corresponding mPDF show no significant changes across $T_f$, indicating that the dynamically fluctuating short-range spin correlations in the paramagnetic state retain the same basic type-III configuration; the only change apparent from the neutron scattering data is a gradual reduction of the correlation length and locally ordered moment with increasing temperature. Taken together, these results paint a unique and detailed picture of the local magnetic structure and dynamics in Zn$_{0.5}$Mn$_{0.5}$Te and show that this material is best described as a cluster spin glass. In addition, this work showcases a statistical method for extracting diffuse scattering signals from neutron powder diffraction data.